The first two sum rules were derived by Weinberg  [1]}, while the third one has been added by the Lastly, Kapusta and Shuryak [2]}.
In the latter sum rule \(O_4^{\mathrm {SB}}\)  stands for the chirally odd combination of four-quark condensates in the vector and the axial channels.
For example, using the first sum rule and assuming \(\delta \)  function for the \(\rho \)  meson, the relation between the \(a_1\)  and \(\rho \)  masses \(m_{a1}=\sqrt{2} m_{\rho }\)  has been derived  [1]}.
As already pointed-out above, the spectral distributions of the vector and axial-vector states are very well known in vacuum from the \(\tau \)  decays into odd (even) number of pions \(\tau \rightarrow \nu _{\tau }(2n+1)(2n) \pi \) .
They provide the reference for the study of the QCDSR and WSR in vacuum and were shown to be satisfied within a few percent if, in addition to the dominant \(\rho \)  and \(a_1\) , contributions from excited states (\(a_1^{\prime }\) , \(\rho ^{\prime }\) ) and the continuum are included in the respective vector and axial-vector spectral distributions [4]}.
Similar studies were also performed in [5]} and [6]}.
